1.1.12 problem 14

There are $4$ choices for sizes and $8$ choices for toppings, of which any combination (including no toppings) can be selected.

The total number of possible choices of toppings is ${\sum }_{i=0}^8\left(\frac{8}{i}\right)=2^8=256$. Thus, the total number of possible size-topping combinations is $4\ast 256=1024$.

We wish to sample two pizzas, with replacement, out of the $1024$ possibilities. By Einstein-Bose, there are a total of $\left(\frac{1025}{2}\right)$ choices.

A common mistake is to use multiplication rule to get $\left(2^8\right)\ast \left(2^8\right)$ as total possible combinations for two pizzas, and try to adjust for overcounting by dividing the result with 2 (as order between pizzas doesn’t matter). This fails because the possibilities with identical pizzas are counted only once.